Research Proposal

Jordan Triple Higher (σ,τ)-Homomorphisms on Prime Rings

October 2019

Authors:

Neshtiman N. Sulaiman

Salahaddin University-Erbil

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ResearchGate has not been able to resolve any references for this publication.

Article

A basic functional identity with applications to Jordan σ-biderivations

April 2017 · Communications in Algebra

Let R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Qml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B: R × R → Qml(R) satisfying [B(x, y), [x, y]] = 0 for all x, y ∈ R must be of the form (x, y) → λ[x, y] + μ(x, y) for x, y ∈ R, where λ ∈C and μ: R × R ... [Show full abstract] → C is a bi-additive map. As applications to the theorem, Jordan σ-biderivations with σ an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.

Article

Full-text available

Multiplicative Jordan triple higher semi-derivations on rings and standard operator algebras

March 2024 · Annals of Mathematics and Computer Science

In this paper we study the additivity of multiplicative Jordan triple higher semi-derivations on rings and standard operator algebras.

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Article

Composition of generalized derivations acting as a Jordan product in prime rings

March 2024 · Communications in Algebra

Chapter

Commutativity of $$ -Prime Rings with Generalized Derivations on $$ -Jordan Ideals

December 2022

The purpose of this paper is to find commutativity conditions in

*

-prime rings with generalized derivations, where

'*'

is involution of the second kind. More specifically, it is shown that if a 2-torsion free

*

-prime ring

\mathscr {R}

with involution of the second kind satisfies any of the following assertions: (i)

[\mathscr {F}(\alpha ),\alpha ^{*}]\in Z(\mathscr {R}),

(ii) ... [Show full abstract]

\mathscr {F}(\alpha )\circ (\alpha ^{*}) \in Z(\mathscr {R}),

(iii)

\mathscr {F}(\alpha )\circ \mathfrak {d}(\alpha ^{*})\pm \alpha \circ \alpha ^{*} \in Z(\mathscr {R})

and (iv)

[\mathscr {F}(\alpha ),\mathfrak {d}(\alpha ^{*})]\pm \alpha \circ \alpha ^{*}\in Z(\mathscr {R}),

where

\mathscr {F}

is a generalized derivation associated with a derivation

\mathfrak {d}

such that

\mathfrak {d}

is commuting with

*

and

\alpha

varies over a nonzero

*

-Jordan ideal of

\mathscr {R}

, then

\mathscr {R}

is commutative.Keywords

*

-prime ringsGeneralized derivationsInvolution

Article

Bounded Distance Equivalence of Cut-and-Project Sets and Equidecomposability

February 2025 · International Mathematics Research Notices

Sigrid Grepstad

We show that given a lattice

\Gamma \subset \mathbb{R}^{m} \times \mathbb{R}^{n}

, and projections

p_{1}

and

p_{2}

onto

\mathbb{R}^{m}

and

\mathbb{R}^{n}

, respectively, cut-and-project sets obtained using Jordan measurable windows W and

W^{\prime}

\mathbb{R}^{n}

of equal measure are bounded distance equivalent only if W and

W^{\prime}

are equidecomposable, up to measure ... [Show full abstract] zero, by translations in

p_{2}(\Gamma )

. As a consequence, we obtain an explicit description of the bounded distance equivalence classes in the hulls of simple quasicrystals.

Article

Full-text available

On integers n with Jt(n)n

January 1989 · International Journal of Mathematics and Mathematical Sciences

Let Ft be the set of all positive integers n such that Jt(n) < Jt(m) for all m > n, Jt(n) being the Jordan totient function of order t. In this paper, it has been proved that (1) every postive integer d divides infinitely many members of Ft (2) if n and n′ are consecutive members of Ft, n/n′ →1 as n → ∞ in Ft (3) every prime p divides n for all sufficiently large n ε Ft and (4) Log Ft(x) « log1/2 ... [Show full abstract] x where Ft(x) is the number of n ε Ft that n ≤ x.

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Preprint

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Alternative $M_2$ -algebras and {\mit \Gamma}-algebras

December 2022

Recently V.H.L\'opez Sol\'is and I.Shestakov solved an old problem by N.Jacobson on describing of unital alternative algebras containing the

2\times 2

matrix algebra

M_2

as a unital subalgebra. Here we give another description of

M_2

-algebras via the 6-dimensional alternative superalgebra B(4,2) and an auxiliar \mathbf {\mit Z}_2-graded algebra \mit\Gamma. It occurs that the category ... [Show full abstract] of alternative

M_2

-algebras is isomorphic to the category of \mit\Gamma-algebras. For any associative and commutative algebra A, we give a construction of a {\mit \Gamma}-algebra \mit\Gamma(A), which turns to be a Jordan superalgebra; if A is a domain then \mit\Gamma(A) is a prime superalgebra. We describe also the free \mit\Gamma-algebras and construct their bases.

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Article

Outer inheritance in Jordan algebras

January 1999 · Communications in Algebra

Kevin Mccrimmon

In the general structure theory of prime, simple, and division Jordan algebras developed by E.I. Zel’manov and applied in his solution of the Burnside problem, the Jordan classification in characteristic 2 required outer ideals of classical algebras. In this paper we show directly that over any ring of scalars the properties of nondegeneracy, strong primeness, unital simplicity, or divisibility ... [Show full abstract] are inherited by any ample outer ideal. This applies in particular to ample subspaces H 0(A,*) of hermitian elements in associative algebras with involution.

Article

Prime algebras and absolute zero divisors

October 2007 · Mathematics of the USSR-Izvestiya

Sergey Valentinovich Pchelintsev

This paper is devoted to the solution of the problem of absolute zero divisors in the varieties of Jordan algebras and (–1,1)-algebras over a field of characteristic 0. It is shown that in those varieties there exist prime algebras generated by absolute zero divisors. Bibliography: 21 titles.

Article

Quaternary Even Positive Definite Quadratic Forms of Discriminant 4 p

June 1999 · Journal of Number Theory

Wai Kiu Chan

Letp>13 be a prime congruent to 1 modulo 4. Let G be the genus of a quaternary even positive definite Z-lattice of discriminant 4pwhose 2-adic localization has a proper 2-modular Jordan component. We show that the orthogonal group of any lattice from G is generated by −1 and the symmetries with respect to the roots of the lattice. The class number of G is computed. Furthermore, we show that the ... [Show full abstract] theta series of degree two coming from the classes in G with non-trivial automorphism groups are linearly independent.

Article

Lie and Jordan structures in simple associative rings

November 1961 · Bulletin of the American Mathematical Society

I. N. Herstein

1. Introduction. In classifying the simple, finite-dimensional Lie algebras over the field of complex numbers, Cartan and Killing showed that, outside of five isolated examples, these fall into four in finite families. These four infinite families are, roughly speaking, the matrices of trace 0 and the skew-symmetric matrices under some involution de fined for the algebra of matrices. It is ... [Show full abstract] natural to ask whether the simplicity of these sets of matrices, as Lie algebras, is really a consequence of the simplicity of the algebra of all nXn matrices over a field, as an associative ring. We shall see that this is indeed the case. The simplicity of an asso ciative ring, and by this we mean any simple ring, with no chain con ditions assumed (it can even be a radical ring) forces the simplicity or almost-simpli city of certain natural nonassociative structures ob tained from the elements and operations of the forementioned asso ciative ring. However, our prime motivation for undertaking these studies is not the mere desire to generalize these well-known results of Cartan and Killing from the classical case of matrices to arbitrary simple rings. Rather, the foremost reason for investigating these structures is to provide a tool, a set of techniques, an approach, for answering purely associative questions about the associative structure of these simple rings. As we proceed in the exposition and development of the researches carried out to date we shall, from time to time, point out how the nonassociative theorems proved can be exploited to prove theorems about simple rings; theorems in whose statements and conclusions these nonassociative structures play no role. These applications, we feel, are but the first ones that have been made. We hope that these techniques will be used advantageously more and more to prove other theorems. Aside from sketching the use of the results in special cases, we shall give no proofs. These can be found in the appropriate references given.

Article

Full-text available

On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains w...

October 2012 · Ukrainian Mathematical Journal

Let ℂ be the complex plane, let

\bar{\mathbb{C}}=\mathbb{C}\cup \left\{ \infty \right\}

, let G ⊂ ℂ be a finite Jordan domain with 0 ∈ G; let L := ∂G; let Ω :=

\bar{\mathbb{C}}\backslash \bar{G}

, and let w = φ(z) be a conformal mapping of G onto a disk B(0, ρ 0) :=

\left\{ {w:\left| w \right|<{\rho_0}} \right\}

normalized by the conditions φ(z) = 0 and $ {\varphi}^{\prime}(0)=1 ... [Show full abstract] $, where ρ 0 = ρ 0(0, G) is the conformal radius of G with respect to 0. Let

{\varphi_p}(z):=\int\limits_0^z {{{{\left[ {\varphi^{\prime}\left( \zeta \right)} \right]}}^{2/p }}d\zeta }

and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G, 0) that minimizes the integral

\iint\limits_G {{{{\left| {{{{\varphi^{\prime}}}_p}(z)-{{{P^{\prime}}}_n}(z)} \right|}}^p}d{\sigma_z}}

in the class of all polynomials of degree deg Pn ≤ n such that Pn(0) = 0 and

{{P^{\prime}}_n}(0)=1

. We study the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to φ p (z) on

\bar{G}

with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain improved estimates for the rate of convergence in these domains.

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Jordan Triple Higher (σ,τ)-Homomorphisms on Prime Rings

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